The Alekseev–Gröbner formula, or nonlinear variation-of-constants formula, is a generalization of the linear variation of constants formula which was proven independently by Wolfgang Gröbner in 1960[1] and Vladimir Mikhailovich Alekseev in 1961.[2] It expresses the global error of a perturbation in terms of the local error and has many applications for studying perturbations of ordinary differential equations.[3]
Formulation
Let be a natural number, let be a positive real number, and let be a function which is continuous on the time interval and continuously differentiable on the -dimensional space . Let , be a continuous solution of the integral equation
The Itô–Alekseev–Gröbner formula
The Itô–Alekseev–Gröbner formula[4] is a generalization of the Alekseev–Gröbner formula which states in the deterministic case, that for a continuously differentiable function it holds that
References
- ^ Gröbner, Wolfgang (1960). Die Lie-Reihen und Ihre Anwendungen. Berlin: VEB Deutscher Verlag der Wissenschaften.
- ^ Alekseev, V. "An estimate for the perturbations of the solution of ordinary differential equations (Russian)". Vestn. Mosk. Univ., Ser. I, Math. Meh. 2, 1961.
- ^ Iserles, A. (2009). A first course in the numerical analysis of differential equations (second ed.). Cambridge: Cambridge Texts in Applied Mathematics, Cambridge University Press.
- ^ Hudde, A.; Hutzenthaler, M.; Jentzen, A.; Mazzonetto, S. (2018). "On the Itô-Alekseev-Gröbner formula for stochastic differential equations". arXiv:1812.09857 [math.PR].